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We consider a $C^1-$semilinear elliptic optimal control problem possiblysubject to control and/or state constraints. Generalizing previous work weprovide a condition which guarantees that a solution of the necessary firstorder conditions is a global minimum. A similiar result also holds at thediscrete level where the corresponding condition can be evaluated explicitly.Our investigations are motivated by G\"unter Leugering, who raised the questionwhether our previous results can be extended to the nonlinearity$\phi(s)=s|s|$. We develop a corresponding analysis and present severalnumerical test examples demonstrating its usefulness in practice.

Sufficient conditions for unique global solutions in optimal control of semilinear equations with $C^1-$nonlinearity